Reaction rate of a composite core-shell nanoreactor with multiple, spatially distributed embedded nano-catalysts

In recent years, metallic nanoparticles have emerged as potent catalysts for various applications Pushkarev et al. (2012); Zhang et al. (2012). In particular, the discovery that gold becomes a catalyst when divided to the nanophase has led to an intense research in this field Haruta (2003); Hutchings and Haruta (2005). In many cases the synthesis and the catalytic applications must be handled in the liquid phase, mostly in water. Secure handling of nanoparticles in a liquid phase can be achieved by polymeric carriers that have typical dimensions in the colloidal domain. Examples thereof include dendrimers Crooks et al. (2001); Noh and Meijboom (2015) or spherical polyelectrolyte brushes Ballauff (2007). Such systems allow one to generate nanoparticles in aqueous phase in a well-defined manner and handle them securely in catalytic reactions.
More recently, thermosensitive colloidal microgels have been used as carriers for metallic nanoparticles in catalysis Lu and Ballauff (2011). Fig. 1 displays the scheme of such a carrier system that may be regarded as a nanoreactor: a thermosensitive network composed of cross-linked chains of poly(N-isopropylacrylamide) (PNIPAM) has been attached to a solid core made of an inert material as, e.g., polystyrene or silica Lu et al. (2006). Metal nanoparticles are embedded in the network which is fully swollen in cold water. Raising the temperature above the critical temperature (32^∘ C for PNIPAM), a volume transition takes place within the network and most of the water is expelled Lu and Ballauff (2011). Lu et al. Lu et al. (2006) have been the first to show that the catalytic activity of the embedded nanoparticles is decreased when shrinking the network by raising the temperature. This effect has been explained by an increased diffusional resistance mass transport within the shrunk network Lu et al. (2006); Lu and Ballauff (2011). A similar model has been advanced by Carregal-Romero et al. when considering the catalytic activity of a single gold nanoparticle embedded concentrically in a PNIPAM-network Carregal-Romero et al. (2010).
Recently, we have shown that the mobility of reactants is not the only important factor: an even larger role is played by the change of polarity of the network when considering mass transport from bulk to the catalyst(s) through such medium Wu et al. (2012); Angioletti-Uberti et al. (2015). This theory is based on the well-known seminal paper by Debye Debye (1942) and considers a single nanoparticle located in the center of a hollow thermosensitive network Angioletti-Uberti et al. (2015). Here, the substrate that reacts at the surface of the nanoparticle diffuses through a free-energy landscape created by the hydrogel environment. In other words, the reactants experience a change in the solvation free energy when entering the gels from bulk solvent, which can be equally regarded as adsorption free energy or transfer free energy. For instance, the free energy of a substrate may be lowered upon entering the network. In this way the number of substrate molecules in the network will be augmented, so that their increased concentration in the vicinity of the catalyst will lead to a higher reaction rate. The free-energy change $\Delta G\stackrel{{\scriptstyle\rm def}}{{=}}G_{\rm in}-G_{\rm out}$ for the substrate outside and inside the network leads to a Nernst distribution for the substrate’s concentration within the system. This effect offers a new way to manipulate the catalytic activity and selectivity of metallic nanoparticles Wu et al. (2012).
In this paper we formulate a more general theory, that is able to account for the geometry of core-shell nanoreactors featuring many catalysts, as shown schematically in Fig. 1. Here, a given number $N$ of catalytic centers are encapsulated randomly in a network. We calculate the total rate of the catalytic reaction for a prescribed geometry of the catalysts, given values of $N$ and $\Delta G$ and specified diffusion constants for the substrate in the bulk and in the network. The rate constant computed in this way can be compared to that characterizing an equal number of particles suspended freely in solution. The present model has been designed to describe the well-studied core-shell systems Lu et al. (2006); Lu and Ballauff (2011), but is equally adapted to the study of systems where catalytic centers are embedded in homogeneous microgels Shi et al. (2014). Up to now, most of the experimental work has been done using the reduction of 4-nitrophenol by borohydride ions in aqueous solution Aditya et al. (2015); Zhao et al. (2015). This reaction can be regarded as a model reaction Herves et al. (2012), since it can be monitored with high precision thus leading to very accurate kinetic data. The rate-determining step proceeds at the surface of the nanoparticles and the mechanism is known Gu et al. (2014). The present theory, however, comprises also the nanoreactors in which enzymes are used as catalytic centers embedded in the network shell Welsch et al. (2012).
In general, diffusion-influenced reactions (DIR) are ubiquitous in many contexts in physics, chemistry and biology Calef and Deutch (1983); Rice (1985); Szabo (1989); Zhou et al. (2008). However, while the mathematical foundations for the description of DIR in simple systems have been laid nearly a century ago von Smoluchowski (1916); Debye (1942); Collins and Kimball (1949), many important present-day problems, including the catalytic activity of composite core-shell nanoreactors, require considering complex geometries and multi-connected reactive boundary systems. The first attempts to consider DIR featuring many competing sinks date back to the 1970s Deutch et al. (1976); Felderhof and Deutch (1976), while more sophisticated methods have been developed subsequently to deal with arbitrary systems comprising many partially reactive boundaries Traytak (1992, 2003); Gordeliy et al. (2009). Along similar lines, the theory developed in this paper, based on general results proved in Ref. Traytak (2003), provides a novel, accurate description of DIR occurring between a small substrate molecule and the catalytic centers embedded in a large, composite nanoreactor system.
Our theory is fully general, in that it covers the whole spectrum of rate-limiting steps in catalysis, from reaction-limited to diffusion-limited reactions. While the theory allows one to compute the reaction rate for an arbitrary catalytic surface turnover rate, closed-form analytical expression are derived for strongly reaction-limited and diffusion-limited reactions. In the limit of a dilute random distribution of NPs encapsulated in a thick hydrogel shell, we find that the overall diffusion-controlled rate constant of our core-shell composites is described by a Langmuir-like isotherm of the form

where $N$ is the number of nanoparticles, $\zeta=D_{i}/D_{o}$ is the ratio of the diffusion constants in the hydrogel ($i$ for inner) and bulk ($o$ for outer), $\varepsilon=a/R_{0}\ll 1$ is the ratio of NP size (radius) to the nanoreactor size and $k_{S}=4\pi D_{o}R_{0}$ is the Smoluchowski rate constant for the nanoreactor as a whole, i.e. the total flux (in units of bulk substrate concentration) of substrate molecules to a stationary perfectly absorbing sink of size $R_{0}$ in the bulk. The above expression is valid for small sizes of the central core. Interestingly, for configurations where the core size becomes of the same order of the whole composite (thin shell), our theory shows that in general the rate constant is increased, up to 40 %, depending on the transfer free energy jump and on the reactant mobility in the shell.
In the limit of slow surface substrate-product conversion rate, i.e. for reaction-limited kinetics, we find that

where $k^{\ast}$ is the intrinsic turnover rate constant that describes the transformation of substrate to product molecules at the nanocatalyst surface (units of inverse concentration times inverse time). This means that when the surface substrate conversion rate constant is weak, the geometrical features of the overall assembly and the mobility of substrate molecules within the hydrogel shell become immaterial. In this case, the crucial control parameter is the transfer free energy jump. The paper is organized as follows. In section II we describe our mathematical model and pose the associated boundary-value problem. In section III, we describe concisely the procedure that leads us to the exact solution of the posed problem (the mathematical details can be found in the appendix). Section IV illustrates an analytic approximation that provides an extremely good description of the exact solution for small core sizes in the physically relevant range of parameters. In particular, we discuss how this formula can be used to derive practical criteria to design nanoreactors with optimized performances. Finally, we wrap up our main results in section V.

We model a core-shell nanoreactor consisting of a polystyrene (PS) core surrounded by a microgel layer as two concentric spheres centered at the origin of a Cartesian 3$D$ frame, as depicted in Fig. 1. We denote with $R_{S}$ and $R_{0}$ the core and shell radius, respectively. The shell is assumed to be a homogeneous continuum, carrying $N$ small nano-catalysts (metal nanoparticles or enzymes) that we model as spheres of radius $a$. For the sake of simplicity, and in accordance to our general multi-sink theory Galanti et al. (2016), we label the PS core as the inner sphere with $\alpha=1$ and position vector $\mbox{\boldmath$L$}_{1}=0$ and denote the position of the $N$ nanocatalysts with the vectors $\mbox{\boldmath$L$}_{\alpha}$, $\alpha=2,3,\dots,N+1$. We want to compute the total reaction rate constant for reactions where a substrate (or ligand) molecule is converted to some product species at the surface of the catalyst spheres. These are endowed with a surface rate constant $k^{\ast}$, which is in general a function of temperature due to underlying thermally activated surfaces processes. Let us denote with $\mathcal{S}_{0}\equiv\{r_{0},\theta_{0},\varphi_{0}\}$ the reference frame with the origin at the nanoreactor center and with $\mathcal{S}_{\alpha}\equiv\{r_{\alpha},\theta_{\alpha},\varphi_{\alpha}\}$ the $N$ reference frames with the origins at the nanospheres centers and the axes parallel to $\mathcal{S}_{0}$ (of course $\mathcal{S}_{1}\equiv\mathcal{S}_{0}$). This formally defines the following $3D$ domains

where $\Omega_{1}=\{|\mbox{\boldmath$r$}_{0}|<R_{S}\}$ denotes the interior of the PS core and $\Omega_{\alpha}=\{|\mbox{\boldmath$r$}_{\alpha}|=|\mbox{\boldmath$r$}_{0}-\mbox{\boldmath$L$}_{\alpha}|<a\}$, $\alpha=2,3,\dots,N+1$, denote the interior of the $\alpha$-th nanosphere. The reactant diffuses with diffusion coefficients $D_{i}$ and $D_{o}$ inside the microgel shell and in the bulk, respectively. In general one can assume $D_{i}<D_{o}$ due to obstructed or hindered diffusion in the microgel Cukier (1984).

We look for solutions for the stationary density of reactants in the bulk and in the microgel as linear combinations of regular and irregular harmonics. Given the multi-connected structure of the boundary manifold $\cup_{\alpha}\Omega_{\alpha}$, we must consider as many local Cartesian reference frames as there are non-concentric boundaries. Thus, we can look for solutions in the form

where $Y_{mn}(\mbox{\boldmath$r$})$ are spherical harmonics, $\xi_{0}=r_{0}/R_{0}$, $\xi_{1}=r_{0}/R_{S}$, $\xi_{\alpha}=r_{\alpha}/a$ for $\alpha=2,3,\dots,N$ and $A_{mn},B^{\alpha}_{mn}$ and $E_{mn}$ are $N+3$ infinite-dimensional sets of unknown coefficients that can be determined by imposing the boundary conditions (17b) and (17d) and the pseudo-continuity conditions at the microgel-solvent interface, eqs (17e) and (17f). This can be done straightforwardly using known addition theorems for spherical harmonics, which results in an infinite-dimensional linear system of equations for the unknown coefficients (see appendix A for the details). Furthermore, making use of known properties of solid spherical harmonics, it is easy to see that the rate constant defined by Eq. (15) is simply given by

As shown in the appendix A, the exact solution to the steady-state problem (17) can be worked out in principle to any desired precision by keeping an appropriate number of multipoles. Remarkably, a simple yet accurate analytical expression can be easily obtained in the monopole approximation (MOA), which corresponds to keeping only the $\ell=0$ term in the multipole expansions (19) Deutch et al. (1976); Felderhof and Deutch (1976). In particular, it is interesting to compute the rate normalized to the Smoluchowski rate of an isolated sink of the same size as the whole nanoreactor in the bulk, i.e. $k_{S}^{-}=4\pi D_{0}R_{0}$. We obtain (see appendix B for the details)

where we recall that $\zeta=D_{i}/D_{0}$ and $k_{S}^{+}=4\pi D_{i}a$. This is the key analytical result derived in this work, that can be readily employed to predict and optimize the geometry and activity of typical core-shell nanoreactors. The quantity $\langle a/L_{\alpha\beta}\rangle$ stands for the average inverse inter-catalyst separation. This can be computed analytically under the reasonable assumption that spatial correlations in the catalysts configurations are negligible (see appendix B),

where $\gamma=R_{S}/R_{0}$ denotes the fraction of the nanoreactor size occupied by the PS core and $\varepsilon=a/R_{0}$ is the non-dimensional size of each catalyst. We see that, since $\varepsilon\ll 1$, one has $1+\varepsilon/3\lesssim C\lesssim 6(1+\varepsilon)/5$, i.e., $C$ is of the order of unity, $1.005\lesssim C\lesssim 1.217$ (taking $\varepsilon\approx 0.0146$ from experiments Mei et al. (2007)).
In the limit of vanishing surface reactivity of the embedded nano-catalysts it is immediate to show from eq. (21) that

We see that, if the surface substrate conversion rate constant is weak, this becomes the rate-limiting step for the overall rate of the nanoreactor, irrespective of the geometrical features of the assembly and of the mobility properties of the hydrogel shell. In this case, it becomes crucial to control the transfer free energy jump to tune the rate of the composite nanoreactor. Conversely, if the catalytic action exerted by the metal nanoparticles encapsulated in the microgel is fast with respect to diffusion, i.e. $k^{\ast}\gg k_{S}^{+}$, expression (21) can be simplified by taking the limit $k^{\ast}\to\infty$. This yields the expression for the fully diffusion-controlled rate

which we will discuss in depth in the following section. Note that for $N=1$ Eq. (24) coincides with the solution of the Debye-Smoluchowski problem Debye (1942) for a single perfectly absorbing sink located at the center of the shell, with $G(r)=\{\Delta G\ \mbox{for}\ a<r\leq R_{0}\,|\,0\ \mbox{for}\ r>R_{0}\}$.
Formulas (21) and (24) have been derived in the monopole approximation, which means that any reflecting boundaries in the problem are not taken into account. Therefore, these should be used to approximate the rate constant of a composite nanoreactor for small to moderate sizes of the central PS core. In the following sections, we provide a thorough characterization of the rate constant of a composite core-shell nanoreactor, computed exactly by solving Eqs. (31), and we compare it to the approximate MOA analytical expression (24) in the physically relevant diffusion-limited regime ($k^{\ast}\to\infty$).

We now discuss in more detail the essential features of the diffusion-controlled rate in Eq. (24). In the monopole approximation, valid for small to intermediate sizes of the central reflecting (inert) core, the role of the latter only enters indirectly through the spatial average $\left\langle{a}/{L_{\alpha\beta}}\right\rangle=Ca/R_{0}$, with $C\simeq 1$. In the swollen configuration, the central core does not occupy a large fraction of the overall nanoreactor volume, with $\gamma\approx 0.3$ (as taken from the experiments reported in Ref. Mei et al. (2007)). Hence, in this regime we expect that the exact size of the core should not play a significant role for the diffusion-controlled rate for relevant values of the physical parameters, i.e., (weak) attraction to the hydrogel $\Delta G<0$ and decreased internal diffusion $\zeta<1$.

In this paper we have developed a detailed theory to compute the total reaction rate of core-shell nanoreactors with multiple catalysts embedded in the shell. The theory is utterly general and allows one to compute the overall reaction rate to any desired accuracy for ($i$) given configuration, dimension and surface reactivity of the encapsulated nanocatalysts, ($ii$) size of the core and the shell, ($iii$) substrate mobility in the bulk and in the shell and ($iv$) transfer free-energy jump for substrate molecules. Furthermore, we computed analytical expressions in the monopole approximation that provide an excellent interpolation of the exact solution for small to intermediate sizes of the central core in the physically relevant range of parameters, i.e. small size and high dilution of the nanocatalysts. Our formulas supply ready-to-use simple tools that can be employed to interpret and optimize the activity of experimentally realizable nanoreactor systems. This shall be particularly useful to estimate the optimal number of embedded NPs, that should reflect a compromise between a resource-friendly design and the highest possible catalytic output. Our analytical treatment predicts an optimal number of NPs given by the following expression

where $f\in[0,1]$ is the desired efficiency, $a$ and $R_{0}$ are the NP and overall nanoreactor sizes, respectively, and $D_{i}$,$D_{o}$ are the substrate mobility in the shell and in the bulk, respectively. For realistic values of these parameters, one gets $N_{1/2}$ of the order of hundreds, a value for which the monopole approximation is still in excellent agreement with the exact solution for core sizes such that $R_{S}/R_{0}\lesssim 0.4$. As discussed already previously, eq. (27) makes it clear that a decisive factor in the design of optimized hydrogel-based nanoreactors must be the tuning of the reactant-hydrogel interaction towards attraction ($\Delta G<0$) for a specific reaction (or mix of reactions). Furthermore, as hydrogel that cause strongly reduced substrate mobility also demand more NPs to achieve high efficiency ($N_{f}\propto D_{o}/D_{i}$), the choice of the shell hydrogel should be made so as to privilege smooth longer-ranged interactions (like electrostatic, hydrophobic, or dispersion) with respect to short-ranged ones (like H-bonds), in order to avoid too sticky interactions that would slow down the reactant mobility substantially due to activated hopping.
Our analytical treatment breaks down if one wishes to push the nanoreactor performances towards full efficiency ($f\to 1$), where the loading number of NP increases rapidly, or in the case of larger core sizes (of the order of the whole spherical assembly). In such cases, the diffusive interaction between NPs can no longer be neglected, as well as the effect of the inert PS core, as diffusive and screening interactions among the different boundaries become important. As a consequence, the full exact solution should be employed to investigate the behavior of the rate and elaborate an optimal design of the composite nanoreactor. Interestingly, we have shown that, as a general situation, increasing both the core and the nanocatalyst sizes either has a rather mild effect on the overall performances, or, more generally, causes a rate-boosting effect, with an increase of the overall rate constant of up to 40 % for values of the core size $R_{S}/R_{0}\gtrsim 0.7$.

S. A-U acknowledges financial support from the Beijing Municipal Government Innovation Center for Soft Matter Science and Engineering. J. D. acknowledges funding by the ERC (European Research Council) Consolidator Grant with project number 646659–NANOREACTOR. M. G. and F. P. would like to thank S. D. Traytak for insightful discussions. F. P. and D. F. acknowledge funding by the CNRS (Centre National de la Recherche Scientifique) under the PICS scheme.